Note: I posted this originally at www.talkphilosophy.orgIMathematics "is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation." However, note the word axiomatically, what is an axiom? It is said to have been something "self explanatory", yet unprovable!No one can prove "A=A" because it requires perception to find "A" which interprets raw data, i.e. reality, to be "A". So, is it not possible that "B" is "A"? Or for that matter that nothing is "A"? Absolutely is it possible, as it is interpreted that way.
IIHow can we prove math? How can we ever prove 1+1=2? Allow me to use an illustration to prove it: suppose I am at an apple orchard. I find an apple, then I find another: how many apples do I have? All I know is that I have one and another, it takes a leap of logic to assert that it is two apples. It requires a matter of belief that 1+1=2. It is likewise with all math, since it is based on previously "proven" statements.
IIINeo-mathmaticians do not seriously contest the authority of the relics of math, nor for that matter its doctrines. This is faith, according to Webster's dictionary[1], in the supernatural. By supernatural, it is meant something "of or relating to an order of existence beyond the...universe". In this universe, 1+1 cannot be proven to equal two without a leap of logic. Therefore 2 does not exist in this universe if it requires the addition of one and another. Math then qualifies as the supernatural.
IVReligion is by definition consisting of two things: that which requires faith, and that which "services" the supernatural. Well, math requires faith to do anything! And since it is part of the supernatural, any practice thereof would be practices to the supernatural. And by practicing math, one thereby services it, or rather services the supernatural. We can then deduce that math is then a religion.

Questions, comments, snide remarks? I am going to be revising it in the next few days, if I do anything to it I will post it here.

I'm not sure if I understand what you're trying to say, but I think you should look into Kurt Godel's Incompleteness Theorem

As far as math requiring faith to do anything, just the act of living requires faith to do anything. You need to have faith that the floor will continue to support you, that gravity will continue to pull you toward the earth, etc.

Personally, I see anything that determines how you usually act as a religion. You don't need to believe in God or any form of a deity.

I think you're both doing violence to the usual senses of the word "faith" and "religion." Both of these words refer to belief of a particular sort.

It takes neither faith nor religion to believe that if I leap in front of a moving bus I'm going to experience some discomfort. I have lots of experience interacting with other rapidly moving things (little sisters, baseballs, on one memorable occasion, a bicyclist). I know people who've been hit by cars. So I think it's reasonable to avoid stepping in front of moving vehicles, even if I can't prove it syllogistically and in Platonic purity.

On the other hand, relgion and faith (in the religious sense) are special because they are a belief about matters not amenable to empirical verification in the usual ways. I don't have a time machine to travel back to the beginning of time to see who did what, nor to visit the end of time. In many monotheisms at least, believing without conclusive proof is precisely the point. That is faith.

Mathematics is just a tool for describing things in a particularly obsessive way. But the things being described already exist. You have to have axioms, or you have nothing to say. I'm not a naive realist - I'm not saying there's something platonically true about F=ma; I'm saying that's a useful way to describe something, the same way, "grab the umbrella, it's started to rain" is. Mathematics is simply a tool. I'm not sure we should freak out about how mathematics falls apart in the quest for the Ultimate Proof, any more than we need to freak out about the unsoundness of the Hammer Religion when we discover hammers are no good for painting with. That's just a misunderstanding of the nature of the tool.

We should agree on the definitions of faith and religion.
Religion: a cause, principle, or system of beliefs held to with ardor and faith. 2) the service and worship of God or the supernatural

Faith: belief in the traditional doctrines of a religion (1) : firm belief in something for which there is no proof (2) : complete trust

One must have complete trust in, our example, that 1+1=2. There could have been a misperception, it could have been nothing added to another (real) thing, it would be perceived as 2, but actually 1.

Moreover, when one counts, one makes a leap to say "1, 2, ..." because 2 is a new concept introduced. It consists of 1+1, but how can we know that one or the other exists independent of perception? It then requires faith.

When one does math problems, it is a practice of a thing which cannot be empirically justified(or for that matter objectively observed) in this universe by man. This qualifies it as the supernatural.

However, with all of these factors combines it becomes a religion. Huzzah!

Mathematics is just a tool for describing things in a particularly obsessive way. But the things being described already exist.

But how would we know? We perceive reality, and misperceive it, there could be a thing which we think exists but doesn't.

You have to have axioms, or you have nothing to say.

Why? We must question every axiom, regardless of how "self explanatory" it might appear.

Karl wrote:One must have complete trust in, our example, that 1+1=2. There could have been a misperception, it could have been nothing added to another (real) thing, it would be perceived as 2, but actually 1.

Moreover, when one counts, one makes a leap to say "1, 2, ..." because 2 is a new concept introduced. It consists of 1+1, but how can we know that one or the other exists independent of perception? It then requires faith.

I have no idea what you're saying here, I'm afraid.

When one does math problems, it is a practice of a thing which cannot be empirically justified(or for that matter objectively observed) in this universe by man. This qualifies it as the supernatural.

It absolutely does not. Let's say I take it into my head that eating brown rice will make me grow a moustache. Somehow I've gotten the idea brown rice will increase my testosterone or some such. There's nothing supernatural about idea at all. It's simply false. Something unproven is simply something unproven. It might be supernatural, but it hardly follows that all unproven things are supernatural.

Also, I find your use of the word "empirically" a bit jarring. You're advocating a view which discounts empiricism utterly, since that's the basis for the axioms about which you have such doubts.

However, with all of these factors combines it becomes a religion. Huzzah!

This also doesn't follow. A belief in ghosts, say, or ley lines, isn't necessarily religious, though might be.

Mathematics is just a tool for describing things in a particularly obsessive way. But the things being described already exist.

But how would we know? We perceive reality, and misperceive it, there could be a thing which we think exists but doesn't.

Repeated testing. This isn't a proof, it's evidence. For those of us not Platonists, no formal proof is necessary. I do not spend my time at the keyboard wracked by the worry that it's going to turn into a rabid attack weasel. It's never happened before, I've never heard of it happening, but I'm unable to prove it cannot. I don't think this undermines everything I think I know.

You have to have axioms, or you have nothing to say.

Why? We must question every axiom, regardless of how "self explanatory" it might appear.

Again, your question doesn't seem to me to follow from my statement. I'm merely saying you need axioms for proofs or mathematics to work on. I'm certainly not suggesting we accept every axiom that comes along at face value.

Karl wrote:Faith: belief in the traditional doctrines of a religion (1) : firm belief in something for which there is no proof (2) : complete trust

One must have complete trust in, our example, that 1+1=2. There could have been a misperception, it could have been nothing added to another (real) thing, it would be perceived as 2, but actually 1.

I don't agree that you have to have complete trust. I am reasonably certain that 1+1=2. In fact I am more certain about that than I am about almost anything else I know. But do I have complete trust or complete belief? Have I eliminated all doubt? Well, no, perhaps I'm living in a world of hallucinations. It's possible. But it's unlikely.

In your definition list, along with faith and religion, maybe you can add a definition of reasonable certainty.

You have to have axioms, or you have nothing to say.

Why? We must question every axiom, regardless of how "self explanatory" it might appear.

An axiom is not an assertion of religious faith. An axiom is not a dogma. If you identify an idea as an axiom, it means you will accept it as truth for now, and base your further conclusions on that axiom. It also implies that, for some reason, the idea cannot be derived from any other axioms you have already identified.

Axioms are defined not because we believe them wholeheartedly, but because they are useful as a basis for deriving other truths. Ideally, one tries to identify the smallest possible number of ideas as axioms. Euclid identified only five axioms as the basis for his geometry.

I don't think you need complete trust or faith to think that 1+1=2. It's just like saying 'I believe in gravity' - no one believes in gravity or has faith in it, you know with certainty that it exists; people can also know things with certainty that might be wrong I suppose, but I still think there's a difference. If you are certain something exists then it's different from just believing it, even if you do believe it to be 100% true. 1+1=2 is always true, it's a principle of maths we set down. You mention that in some cases maybe we are decieved into thinking that two things have been added to each other when actually nothing has been added at all. I don't think that 1+1=2 can be applied to things going on around us, eg two rain drops added together are one big raindrop, but that doesn't mean that 1+1=2 is something you can doubt. That equation is one of the rules needed for maths to work and within our system of maths everything must be based on those axioms. You can doubt wether two raindropps will really equal 2 raindropps when you add them together in reality, but you can't doubt that 1+1=2 in our system of mathmatics.

Emma_85 wrote:...you can't doubt that 1+1=2 in our system of mathmatics.

Of course you can't, in our system of mathematics. You can, however, doubt the system itself. Our system of mathematics defines a bunch of rules that determine how numbers behave. Just a bunch of rules, defined by a bunch of people. The fact that 1+1=2 is no more inherent to reality than the fact that success=money. Everything is just a conceptualization of the chaos that is reality.

the Principia Discordia wrote:With our concept making apparatus called "mind" we look at reality through the ideas-about-reality which our cultures give us. The ideas-about- reality are mistakenly labeled "reality" and unenlightened people are forever perplexed by the fact that other people, especially other cultures, see "reality" differently. It is only the ideas-about-reality which differ. Real (capital-T True) reality is a level deeper that is the level of concept.

We look at the world through windows on which have been drawn grids (concepts). Different philosophies use different grids.

A culture is a group of people with rather similar grids. Through a window we view chaos, and relate it to the points on our grid, and thereby understand it.

Mathematics is simply another grid, as are all religions. To say that mathematics is just another grid is to say that it is a set of conceptualizations used to look at reality. I would consider that a religion, but I can see why some wouldn't.

Of course you can't, in our system of mathematics. You can, however, doubt the system itself. Our system of mathematics defines a bunch of rules that determine how numbers behave. Just a bunch of rules, defined by a bunch of people. The fact that 1+1=2 is no more inherent to reality than the fact that success=money. Everything is just a conceptualization of the chaos that is reality.

Hehehe, that's what I (tried to) say in another post on maths. I think maths is made by us humans and is so linked to our reality, how we see the world. I'm not saying humans are born with maths, but something like that anyway. Uh... I need to think about my reply... be back after dinner...

Ok, new try... (I'll sort out my thoughts while i type, so this is probably a mess) I think there are some things more linked to reality or at least to our human existance than other things we believe. Some things we are forced to think of as true by nature (that if we pick up an apple that it will fall for example). We don't believe it will fall, we know. The feeling we have ok knowing and believing something is different (otherwise why have different words for the feeling ? ). 1+1=2 falls under the knowing category not the believing category, that I know , so the question I have to ask myself why does it fall into that category, why isn't it something you can choose or not choose to believe? I think it must have something to do with how our brains work, a quantiy filter in our brain. If I look at a large pile of say 8 apples and next to it a small pile of only 3 without even thinking about maths I'll automatically know which pile has more apples in it. My brain doesn't need maths to know there are more of one thing here than there are over there. If the small pile is mine and the larger my sisters my brain will also tell me that I can make mine larger my adding to mine . It's not really possible to believe or not to believe in something, when our nature tells us that it's a truth we should not question.
I can't see Gods anywhere though, they don't feature in reality, which is why I have to believe in a belief system, the feeling is a different one.
I realise that our system of maths may not be an adequate system, but it's based on our nature. Because we are humans we can't believe it to be a good system or not, it's the only system we can really understand, a really strange alien from another universe might say that he believes our system is good or not, just like he can believe in a god or not, but we humans have a different feeling when we talk about 1+1=2.
Hope that some of it made sense...

Titus Marius Crispus wrote:Of course you can't, in our system of mathematics. You can, however, doubt the system itself. Our system of mathematics defines a bunch of rules that determine how numbers behave. Just a bunch of rules, defined by a bunch of people. The fact that 1+1=2 is no more inherent to reality than the fact that success=money. Everything is just a conceptualization of the chaos that is reality.

The problem here is our system of mathematics. Mihi math is math is math. We may have different ways of expressing it, but this 'bunch of rules' business is really beyond the question. Tthese 'rules' you refer to are really the discoveries of people that help us understand the world better and make it easier to apply mathematics.

If 1+1 semper = 2, I would say that's inherent to reality, wouldn't you? I can think of no case in which 1+1 = 1 or 1=4, unless you want to get witty and come up with something that I bet I can find a flaw in.

I think the fact that 1+1=2 is just a matter of considering ressemblances. In the example with the apples, you consider that the 2 apples both belong to the class of apples, undistinctely. So you have 2 members belonging to the same structure, so you may add them up. In reality, the two apples are not the same; one is bigger than the other, or more red than green, or whatever differences you may find in its characteristics. Actually, if you consider it in a really puristic way, when you have two apples from an orchard you do NOT have 1+1 = 2 apples, but you just have 1 apple + 1 apple, since they are different. It's just like saying, "3 apples + 2 oranges = What?". Well, it equals 5 fruits, if you want, but by saying so, you put them all in a greater class, that is that of the fruits.
You can actually never have two identical things, you only have separate and distinct items, that you put together into a certain cluster. But this matter corresponds to a subjective way of seing things. For you, the apples may seem identical, but to an expert in fruitology (or whatever you call this subject), they could under no circumstances be brought together.
And finally, you can NEVER have two identical things. The mere fact that you make a distinction between them (saying "I have an apple and another apple") means that they are different - in time, space or characteristics. The Aristotelian logic, first rule, states that A =id. A. And that is correct, since anything is just equal to itself and never ever to anything else. Accordingly, all Maths rely on accepted similarities between different items.